#6443317

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437935
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,13)(8,12)(9,15)(10,16)(11,18)(14,25)(17,29)(19,27)(20,32)(21,26)(22,30)(23,33)(24,31)(28,34)(35,36);(5,7,9,18,10,20,25,19)(6,11,14,13,16,27,15,32)(8,21,23,34,22,35,24,17)(12,28,31,26,30,29,33,36);(1,2,3,4)(5,8)(6,12)(7,17)(9,23)(10,22)(11,26)(13,29)(14,31)(15,33)(16,30)(18,21)(19,35)(20,34)(24,25)(27,36)(28,32);(5,7)(6,13)(8,17)(9,18)(10,20)(11,15)(12,29)(14,27)(16,32)(19,25)(21,23)(22,34)(24,35)(26,33)(28,30)(31,36);(8,22)(12,30)(17,34)(21,35)(23,24)(26,36)(28,29)(31,33)

Polycyclic presentation:

2^9|b^2=f,c^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318643, #72318859
maximal normal subgroup of	#72318643, #72318859

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350513);